# Whether a particular fiber product of varieties is integral

Let $$X,Y,Z$$ be irreducible projective varieties over $$\mathbb{C}$$ (you can assume all of them are normal), and let $$f:X\rightarrow Z, g: Y\rightarrow Z$$ be two birational projective morphisms, such that

• there is a common open set $$U\subset Z$$ with $$f_U:f^{-1}(U)\simeq U$$ and $$g_U:g^{-1}(U)\simeq U$$,
• If $$W:= Z-U$$, then $$f_W: f^{-1}(W)\rightarrow W$$ and $$g_W: g^{-1}(W)\rightarrow W$$ are projective bundles of ranks $$m,n$$ respectively, and
• codim$$(W)=1+m+n$$.

Is it true that $$X\times_{Z}Y$$ is integral? I have seen a statement along this line, but can't prove it.

Any help would be really appreciated.

Edit: Thanks to R. van Dobben de Bruyn's comment, the answer to the question as I stated earlier was false. I had skipped the codimension condition.

• This is false. For example, let $f$ and $g$ both be the blowup $\tilde X \to X$ at a point $p$ in a smooth variety $X$ of dimension $d \geq 3$. Then the exceptional divisor is $\mathbf P^{d-1}$, so the fibre product has a subvariety isomorphic to $\mathbf P^{d-1} \times \mathbf P^{d-1}$ lying over $p$. But $2d-2 > d$, so $\tilde X \times_X \tilde X$ has components of different dimensions! Jul 17, 2020 at 14:24
• @R.vanDobbendeBruyn : Thanks for the answer! Actually, there were more conditions on the codimension of $W$, but I thought that it was not required. I have edited my question. Jul 17, 2020 at 16:54